Mathematics for the interested outsider

Integrals and Diffeomorphisms

Let’s say we have a diffeomorphism from one -dimensional manifold to another. Since is both smooth and has a smooth inverse, we must find that the Jacobian is always invertible; the inverse of at is at . And so — assuming is connected — the sign of the determinant must be constant. That is, is either orientation preserving or orientation-reversing.

Remembering that diffeomorphism is meant to be our idea of what it means for two smooth manifolds to be “equivalent”, this means that is either equivalent to or to . And I say that this equivalence comes out in integrals.

So further, let’s say we have a compactly-supported -form on . We can use to pull back from to . Then I say that

where the positive sign holds if is orientation-preserving and the negative if is orientation-reversing.

In fact, we just have to show the orientation-preserving side, since if is orientation-reversing from to then it’s orientation-preserving from to , and we already know that integrals over are the negatives of those over . Further, we can assume that the support of fits within some singular cube , for if it doesn’t we can chop it up into pieces that do fit into cubes , and similarly chop up into pieces that fit within corresponding singular cubes .

But now it’s easy! If is supported within the image of an orientation-preserving singular cube , then must be supported within , which is also orientation-preserving since both and are, by assumption. Then we find

In this sense we say that integrals are preserved by (orientation-preserving) diffeomorphisms.

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This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.